We consider the prediction of a target’s destination and simultaneously recover its filtered trajectory. Two novel models for trajectories with known destinations are presented using reciprocal stochastic processes and stochastic contextfree grammars. We present a destination-aware syntactic tracker which uses conventional state-space estimates from a legacy tracker to perform prediction and trajectory estimation. We also provide statistical signal processing algorithms for model prediction and maximum likelihood sequence estimation using the proposed trajectory models. Simulation results show that both models considered in the paper have superior estimation performance compared to conventional hidden Markov modeling and can reliably predict the target’s destination.